A067015 -id:A067015 - cp's OEIS frontend

A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

1, 5, 22, 217, 4726, 164258, 14840533, 1834600977, 669877863205, 296979228487760, 434542100979981567, 692625866382651263578, 4053364289624915167879497, 23237986479606982160703729647, 543749373021017146939376423644362, 11213018647250714014261414954480048385

Offset: 1

R. H. Hardin, Jan 25 2002

nonn

			Neighbors for n=4:
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
Neighbors for n=5:
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
J. Katzenelson and R. P. Kurshan, S/R: A Language for Specifying Protocols and Other Coordinating Processes, pp. 286-292 in Proc. IEEE Conf. Comput. Comm., 1986.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]

Cf. A006506, A027683, A066863, A066864, A066866, A067967 (shifted instead of bent).

Row sums of A067015.

More terms from Sean A. Irvine, Nov 18 2023

A377700 Number of ways of placing n nonattacking rooks on a toroidal board of 2n^2 equilateral triangular spaces.

Original entry on oeis.org

2, 4, 6, 48, 30, 1152, 266, 45824, 4050, 2736000, 75702, 233017344, 2060734

Offset: 1

Hugh Robinson, Nov 04 2024

The board is formed by identifying opposite edges of a 60°/120° rhombus, tiled by equilateral triangles with n triangles along each edge.
A rook move on the equilateral triangle tessellation is a move along a path through successively edge-adjacent faces, that turns alternately left and right at each face (starting with either left or right), so that the overall direction of movement remains approximately parallel to one set of triangle edges.
Also counts the number of 3 X n matrices such that each row is a permutation of {1, ..., n}, the first row is the identity permutation (1 .. n), and each column sums to either 0 or 1 (mod n).
When n = 2k+1, the rooks must be either all on white spaces or all on black spaces (using the obvious parity coloring) and the problem is equivalent to A006717(k) for each case. When n = 2k, there are k rooks on white spaces and k rooks on black spaces.

			For n = 4, the a(4) = 48 arrangements are generated from one solution by symmetries of the toroidal grid:
              o---o---o---o---o
             / \ /X\ / \ / \ /
            o---o---o---o---o
           / \X/ \ / \ / \ /
          o---o---o---o---o
         / \ / \ / \ / \X/
        o---o---o---o---o
       / \ / \ /X\ / \ /
      o---o---o---o---o

Cf. A375800, A006717, A002047, A062166, A067015, A342372.

% minizinc -D 'N=6' -s --all-solutions a.mzn
include "globals.mzn";
include "alldifferent.mzn";
int: N;
array[1..N] of var 1..N: perm1;
array[1..N] of var 1..N: perm2;
constraint alldifferent(perm1);
constraint alldifferent(perm2);
constraint forall(i in 1..N)(perm1[i] + perm2[i] + i in {N,N+1,2*N,2*N+1,3*N});
solve satisfy;
output [show(i) ++ " " | i in 1..N];
output [show(perm1[i]) ++ " " | i in 1..N];
output [show(perm2[i]) ++ " " | i in 1..N];

a(2k+1) = 2 * A006717(k).

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A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

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A377700 Number of ways of placing n nonattacking rooks on a toroidal board of 2n^2 equilateral triangular spaces.

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